The Philosophy of Quantum Mechanics: An Interactive Interpretation [1st paperback ed., repr.] 9780521371056, 0521371058, 9780521408745, 0521408741

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The philosophy of quantum mechanics

The philosophy of quantum mechanics An interactive interpretation

Richard Healey

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Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 40 West 20th Street, New York, NY, 10011-4211, USA 10 Stamford Road, Oakleigh, Victoria 3166, Australia © Cambridge University Press 1989 First published 1989 Reprinted 1990 First paperback edition 1990 Reprinted 1991 Library of Congress Cataloging-in-Publication Data

Healey, Richard. The philosophy of quantum mechanics: an interactive interpretation / Richard Healey p. cm. Bibliography: p. ISBN 0-521-37105-8 1. Quantum theory. 2. Physics - Philosophy. 1. Title. QC174.12.H43 1990 89-30943 530.1'2-dcl9 CIP British Library Cataloguing in Publication Data

Healey, Richard The philosophy of quantum mechanics. 1. Quantum theory. I. Title 530.1'2 ISBN 0-521-37105-8 hardback ISBN 0-521-40874-1 paperback

Transferred to digital printing 2003

For Jean

Contents Preface

page xi

Introduction 1 What I take to be an interpretation of quantum mechanics, and why any further interpretation is needed: the chief problems with the most popular existing interpretations (both Copenhagen and nonCopenhagen). 1 Overview 26 Basic ideas of the interpretation to be presented here, together with idealized examples illustrating their application, including repeated Stern-Gerlach "measurements" and coupled spin-Vi systems. 2 Dynamical states 63 Dynamical states as collections of quantum dynamical properties. The structure and evolution of dynamical states. §2.1 The hierarchy of quantum systems 63 How compound quantum systems are composed of their subsystems, and how their dynamical properties are related to those of their components. Prime and composite properties. §2.2 The structure of dynamical states 66 Conditions on properties contained in the dynamical state of a quantum system. How these conditions relate to the hierarchy of quantum systems, and to the presence or absence of interactions. §2.3 Dynamics 79 The dynamical law underlying the Schrodinger equation, and a stability condition governing dynamical states of interacting systems. vn

3 Measurement and quantum states 84 The nature of quantum states: measurement, preparation, and the derivation of the Born probability rules. Dissolution of the measurement problem. §3.1 Measurement interactions 84 Measurements as quantum interactions. Definition of simple Mtype interactions. Mimicking of the projection postulate through verifiability of measurement results. Motivation for the distinction between prime and compound properties. §3.2 Idealizations relaxed 94 How the above account of measurement interactions may be rendered more realistic. §3.3 The assignment of quantum states 104 The character and legitimacy of quantum state ascriptions. Preparations as quantum interactions. Definition of the class of simple P-type interactions. Derivation of the Born rules in special cases. 4 Coupled systems 116 The present interpretation is applied to coupled systems of the type studied by Einstein-Podolsky-Rosen, Bell, and Aspect I: technical details. 5 Metaphysical aspects 137 The present interpretation is applied to coupled systems of the type studied by Einstein-Podolsky-Rosen, Bell, and Aspect II: metaphysical aspects and implications concerning holism, nonseparability, causal explanation, and causation. How quantum mechanics explains the observed properties of these correlated systems according to the present interpretation. 6 Alternatives compared 180 The present interpretation is compared to several existing interpretations in order to clarify its logical and genetic relations to them as well as to point out its advantages. §6.1 Naive realism 180 Comparison with an interpretation often attributed to Einstein. §6.2 Copenhagen 184 Comparison with two prominent versions of the Copenhagen interpretation. viii

§6.3 Everett 205 Comparison with Everett's relative state interpretation, popularly known as the "many worlds" interpretation. §6.4 Kochen 216 Comparison with interpretative views put forward recently by Kochen. 7 Open questions 235 A number of open questions for the present interpretation are posed and discussed. These include the possibility of generalizing the present treatment of measurement and preparation interactions, of extending this interpretation to a relativistic theory, of investigating the nature of the stochastic dynamics underlying quantum probabilities in the present interpretation, and of proving certain important limit results. Appendix 253 Containing proofs of several results and lemmas stated and used in the text. Selected bibliography

261

Index

265

IX

Preface In this book I present a new interpretation of the theory of nonrelativistic quantum mechanics. In this interpretation, measurement in quantum mechanics corresponds to a physical interaction internal to a compound quantum system, which correlates the dynamical states of measured system and (quantum mechanical) apparatus system; whereas the quantum state provides (via the Born rules) a summary of numerical information concerning probabilistic dispositions realized in such interactions. The distinction between dynamical and quantum states is maintained to be the key to the dissolution of the measurement problem. The further idea that the dynamical state of a compound quantum system is not always determined by those of its components then permits a novel understanding of EPR-type correlations as manifested in experiments such as those of Aspect, Grangier, and Roger (1982a) and Aspect, Dalibard, and Roger (1982b). This book is a monograph: Our present understanding of quantum mechanics does not suffice for the composition of a treatise. It arose when the continual frustration of my attempts to understand quantum mechanics in accordance with any existing interpretation forced me to try something new. I have no illusions that this monograph finally renders quantum mechanics transparent and manifestly free of all conceptual problems. But I do believe that new ideas are urgently needed if we are to approach this happy state; and that there is a good chance that the particular combination of ideas contained in the present interpretation represents significant progress toward this goal. Consequently, I have written XI

mostly in a spirit of advocacy for the interpretation presented here, developing the view so that its advantages are emphasized. But I have also tried to clearly acknowledge difficulties, particularly in the concluding chapter. I hope that some reader will be motivated by the promise of the view to develop or modify the interpretation so as to overcome such difficulties. After an introductory chapter which explains and motivates the attempt to provide a new interpretation of quantum mechanics, Chapter 1 outlines the central ideas of the new interpretation. The detailed presentation begins in Chapter 2, where the character of dynamical states is investigated in some depth. Chapter 3 analyzes quantum measurement and preparation interactions, explains the role of the quantum state, and discusses the interpretation and origin of the Born probability rules of quantum mechanics. Chapter 4 gives technical details of the application of the present interpretation of quantum mechanics to coupled systems of the type studied by EPR, Bell, and Aspect, and Chapter 5 explores metaphysical aspects and implications of this application concerning holism and causal explanation. In Chapter 6 I explicitly compare the interpretation to certain other interpretations which were influential in its development: The present interpretation represents an evolution from earlier views, not a radical break with them. I conclude in Chapter 7 by discussing a number of problems still to be faced by the interpretation presented here. A few proofs whose presentation in the text would have interrupted the exposition have been relegated to an appendix. I first began the work which resulted in this monograph in the winter of 1984 while I was a visiting fellow of the Center for Philosophy of Science at the University of Pittsburgh. I wish to thank the National Endowment for the Humanities, whose award made this visit financially possible, and the Center and its Director Nicholas Rescher and secretary Linda Butera, whose generous hospitality, both then and later (durxii

ing the academic year 1987-8), provided me with ideal working conditions in a friendly and stimulating intellectual community. After my return to Los Angeles, continued work on the project during the period 1985 through 1987 was facilitated by the NSF. Consequently, this monograph is, in part, based upon work supported by the National Science Foundation under Grant no. SES8512026: It does not, of course, reflect the views of that organization. I am grateful to the NSF for their support. I have had the benefit of presenting various earlier versions of parts of this work before a number of audiences, including the philosophy of physics mini-conference at Stanford University organized by Nancy Cartwright in February 1985; the philosophy departments of Cornell University, the University of California, Santa Barbara, Arizona State University, Rice University, Columbia University, Amherst College, the University of Western Ontario, Pittsburgh University, and the University of California, Davis; and the Joint US-USSR colloquium on the foundations of quantum mechanics at Easton, Maryland, organized by Jeffrey Bub in September 1988. I wish to single out the Relativity Group at the University of Chicago for special mention. Under the active leadership of Robert Geroch, participants at a talk I gave there in April 1984 not only reacted in a constructive and openminded way to thoughts which at that stage were considerably less than half-baked, but also took pleasure in resolving technical questions in a matter of minutes, a positive answer to which at that time was crucial to the continued development of my interpretation. I have a number of individuals to thank for constructive criticisms of my work during this period, including David Albert, Geoffrey Hellman, David Malament, Hilary Putnam, Abner Shimony, Allen Stairs, and Roberto Torretti. I am grateful to Howard Stein and Al Janis, both of whom read a penultimate draft of the first five chapters of the manuscript with exceptional speed, care, and thoughtfulness, and offered advice which I have usually taken (and ignored at my peril!). xin

Earlier in the project, conversations with both Al and Tony Martin were crucial in determining the direction of my thinking; although I am confident that neither of them would endorse the present interpretation, and certainly neither of them can be held responsible for the consequences of their interventions. My greatest debt is to my family, and especially to my wife, Jean Hampton. As a colleague she has provided constant encouragement and wise philosophical counsel while I have been working on this project. As a friend she has been a source of emotional support during some difficult times. But the countless sacrifices she has made on my behalf, of career opportunities as well as of her own precious time, far exceed those appropriate to any colleague or friend: They constitute a debt which I can never hope to repay, but can only wonder at. Davis, California January 1989

xiv

Introduction In this monograph I present the basic structure of an interpretation of quantum mechanics. Chapter 1 outlines the central ideas behind the interpretation and illustrates them by means of examples: Subsequent chapters fill in the details of the interpretation. But it is important to begin by explaining what I take an interpretation of quantum mechanics to be, and why any further interpretation needs to be offered. After all, quantum mechanics (in some form) is by now both a foundation for much of contemporary physics and a veteran of more than sixty years of intermittent but sometimes intense reflection on its content and meaning. What more can be, or should be, said about the interpretation of this theory? Many physicists believe that no more needs to be said: that there is basically only one way of understanding quantum mechanics, due to Bohr, Heisenberg, Pauli, and others, and that the only remaining interpretative task is that of the physics teacher, who seeks to perfect ways of conveying this understanding to new generations of students. Sir Rudolf Peierls, one of the more lucid and distinguished of these physicists, even objects to the use of the familiar term 'Copenhagen interpretation' to refer to the way of understanding quantum mechanics due to Bohr, Heisenberg, Pauli, and others. Because this sounds as if there were several interpretations of quantum mechanics. There is only one. There is only one way in which you can understand quantum mechanics. There are a number of people who are unhappy about this, and are trying to find something else. But nobody has found anything else which is consistent 1

yet, so when you refer to the Copenhagen interpretation of the mechanics what you really mean is quantum mechanics. And therefore the majority of physicists don't use the term; it's mostly used by philosophers.1 I am a philosopher, and I shall sometimes find it convenient to talk about the Copenhagen interpretation of quantum mechanics. But that may well be the only respect in which I can assent to the views expressed by Peierls in this passage. It seems to me that far from there being only one interpretation of quantum mechanics, there is today no fully satisfactory way of understanding this theory. Instead we are faced with an extraordinary variety of attempts to understand quantum mechanics: Indeed, it sometimes seems as if there are as many different attempts as there are people who have seriously made the attempt! But none of these attempts has either won, or deserved, universal or even widespread acceptance. It is sometimes useful to classify these different interpretationsketches, since they do fall into certain groups. Thus one may refer to the many-worlds interpretation, to an interpretation in terms of hidden variables, to a naive realist interpretation, to the quantum logical interpretation, or to the Copenhagen interpretation. But such references should not be taken to be more definite than they are. There are, for example, many ways of trying to implement the basic ideas behind "the" many-worlds interpretation. And these do not amount to mere stylistic variants: Each of them gives rise to a very different conception of quantum mechanics. This is true also of "the" Copenhagen interpretation. Bohr, Heisenberg, and Pauli each held significantly different views on how quantum mechanics should be understood. And the views of von Neumann and of Wigner diverge even more radically from these, although they consider themselves to be proponents of the very same interpretation, and are often taken at their word. Textbook writers typically pay lip service 'This passage is quoted from the edited transcript of a radio interview, and appears on page 71 of Davies and Brown (1986).

to the ideas of Bohr, but neither state these clearly and accurately, nor use them to develop any coherent understanding of quantum mechanics.2 One might still wonder whether there is some version of "the" Copenhagen interpretation which is preferable to any other, and clearly superior to all non-Copenhagen views. If so, it would lay claim to be Peierls's one true interpretation of quantum mechanics. Though I do not believe that there is any such version, justifying this belief would require an exhaustive study of the many versions that have actually been proposed, as well as a number that could have been proposed. It is not the purpose of this monograph to undertake such a study. Instead, in the second section of this introduction I shall sketch the basic ideas behind two common versions of the Copenhagen interpretation, and outline what I take to be the chief reasons for rejecting them. This is not intended as a definitive refutation of "the" Copenhagen interpretation, but as a preliminary statement of reasons for looking beyond it. I am not alone in my dissatisfaction with what I call the Copenhagen interpretation. Although many physicists at least pay lip service to this interpretation of quantum mechanics, there is a significant, and perhaps growing, minority who reject it in favor of something else. Rivals to Copenhagen orthodoxy now include naive realism,3 (nonlocal) hidden variable theories,4 the Everett, or many-worlds, interpretation,5 and the quantum logical interpretation.6 It is therefore appropriate for me to address the question of the relation of the interpretation to be presented in this mono2

David Bohm's (1951) Quantum Theory is a welcome exception to this generalization. See Ballentine (1970) for one physicist's presentation of this view. It is arguable that Einstein held a naive realist view of quantum mechanics. I have criticized this view in Healey (1979). 4 See, for example, Vigier (1982). 5 For defenses of this view see, for example, DeWitt and Graham (1973), and Geroch (1984). For objections, see, for example, Healey (1984), and Stein (1984). 6 For one physicist's presentation of this view see Finkelstein (1962). The quantum logical interpretation seems more popular among philosophers and mathematicians: see Putnam (1968), Bub (1974), and Friedman and Putnam (1978). 3

graph to these other "unorthodox" approaches. It is clear that my presentation should not be considered part of either the naive realist or the quantum logical traditions. I should myself also resist its assimilation into either the hidden variable or the many-worlds tradition; though others may classify the view differently. The important point is that, in my opinion, there are powerful arguments against the usual versions of all the familiar unorthodox interpretations [though these are not so powerful as Peierls implies in the quoted passage; a quantum logical interpretation might be accused of (classical) inconsistency, but I doubt that any of the other interpretations can be shown to be inconsistent]. I shall sketch some of these usual versions in the second section, and then argue against them. Again, I must stress that I do not take any of these arguments to provide a definitive refutation of the interpretation against which it is offered. That would at least require considerable sympathetic reconstruction of each interpretation, combined with systematic and wide-ranging criticism. My purpose is the more limited one of establishing a prima facie case against each interpretation to motivate my own presentation of still another interpretation in the rest of this monograph. The preceding discussion assumes that it is clear both what an interpretation of quantum mechanics would be and that it is necessary to find an interpretation that is in some sense acceptable. In fact, this is not so clear as to go without comment. Let me start from Peierls's equation of an interpretation of a theory with a way of understanding that theory. Why should it be necessary to provide, in addition to the theory itself, something further, namely a way of understanding the theory? This would be necessary if one took the theory to be an uninterpreted formal system, on the positivist model for a scientific theory.7 In this model, the theory would require 7

See, for example, Carnap (1939).

supplementation by semantic principles in order for its constituent terms and sentences to be endowed with meaning, and there may be controversy as to just what these semantic principles should be. But the general positivist model for a scientific theory has justifiably come under sustained criticism in recent philosophy of science, and there seems little reason to suppose that quantum mechanics conforms to this model better than other theories. Furthermore, disputes about how quantum mechanics should be understood extend down to the level of disagreement over how one would go about formalizing the theory; consequently, there is no agreed formal system whose semantics is in doubt. It is true that there is widespread agreement that quantum mechanics employs certain by now well-understood mathematical structures: For example, in quantum mechanics dynamical variables are representable by selfadjoint operators on a Hilbert space, whose spectra represent possible values of these variables. But such agreement does not extend to the exact wording of the fundamental principles of the theory, nor even to just what these principles are. For example, the status of the projection postulate - von Neumann's process 1 - has long been highly controversial. There are those who have held some form of this principle to be an essential postulate of the theory; others have taken it to hold only in certain special circumstances; still others have considered the principle to be actually inconsistent with the fundamental principles of quantum mechanics. The exact statement of the basic (Born) probability rules has also been a highly controversial matter: Do these specify probabilities that a quantity has a certain value, that it will or would acquire a certain value on measurement, or that a measuring apparatus will or would record a corresponding result if the quantity is measured? In addition to disagreement over the status and exact formulation of such important theoretical principles, there has been no clearly agreed upon understanding of central no-

tions of quantum mechanics such as those of measurement and of quantum states. Here we have an interpretative problem that more closely fits the positivist paradigm: Just what is meant by terms like 'measurement' and 'quantum state' as they figure in quantum mechanics? Finally, it is well known that the conceptual foundations of quantum mechanics have been plagued by a number of "paradoxes," or conceptual puzzles, which have attracted a host of mutually incompatible attempted resolutions - such as that presented by Schrodinger (1935), popularly known as the paradox of Schrodinger's cat, and the EPR "paradox," named after the last initials of its authors, Einstein, Podolsky, and Rosen (1935). A satisfactory interpretation of quantum mechanics would involve several things. It would provide a way of understanding the central notions of the theory which permits a clear and exact statement of its key principles. It would include a demonstration that, with this understanding, quantum mechanics is a consistent, empirically adequate, and explanatorily powerful theory. And it would give a convincing and natural resolution of the "paradoxes." I should like to add a further constraint: that a satisfactory interpretation of quantum mechanics should make it clear what the world would be like if quantum mechanics were true. But this further constraint would not be neutral between different attempted interpretations. There are those, particularly in the Copenhagen tradition, who would reject this further constraint on the grounds that, in their view, quantum mechanics should not be taken to describe (microscopic) reality, but only our intersubjectively communicable experimental observations of it. It would therefore be inappropriate to criticize a proposed interpretation solely on the grounds that it does not meet this last constraint. But this constraint will certainly appeal to philosophical realists, and for them at least it should count in favor of an interpretation if it meets this constraint, as does

the interpretation presented in this monograph - or so I hope to show.8 Granted that quantum mechanics requires some interpretation, why is a new interpretation needed? Why are none of the interpretations so far offered acceptable? Even a preliminary answer to these questions calls for a discussion of the key points of some of the more prominent contemporary approaches to quantum mechanics. It is convenient to introduce this discussion by referring to the Born rules, which are customarily taken to be the central probabilistic principles of quantum mechanics: Different approaches may be distinguished by their different interpretations of the Born rules. Leaving aside joint probabilities, these may be taken to have the following form: prob+(rfEft) = p.

(1.1)

Here p is a real number between zero and one (including those limits), si is a quantum dynamical variable, 11 is a (Borel) set of real numbers, and i|/ is a mathematical representative of an instantaneous quantum state. A preliminary reading of (1.1) is as follows: "In quantum state iji, the probability of finding that the value of si lies in Cl is p." But how is the phrase 'of finding' to be understood? Is this phrase just a redundant rhetorical device inserted to draw attention to the fact that instances of (1.1) are testable by repeatedly measuring the value of si on each of a large number of similar systems in quantum state i|/ and observing in what fraction of the tested cases that value lies in £1? Or would the omission of this phrase constitute a substantive distortion of the content 8

It is interesting to note that its appeal extends also to certain antirealists. Before offering his own interpretation of quantum mechanics, Van Fraassen (1981), certainly no scientific realist, formulates the interpretative task of the philosopher of science as that of "describing how the world can be the way that scientific theories say that it is" (p. 230).

of (1.1), which is intended to apply explicitly to the results of measurements of si, and not to the value si has independent of whether or not it is measured? One main approach to quantum mechanics takes the first option: According to an approach I have elsewhere characterized as naive realist (see Healey, 1979), the Born rules apply directly to possessed values of quantities, and only derivatively to results of measurements of these quantities. According to naive realists every quantum dynamical variable always has a precise real value on any quantum system to which it pertains, and the Born rules simply state the probability for that value to lie in any given interval. Thus, for them, the Born rules assign probabilities to events involving a quantum system a of the form "The value of si on a lies in ft." A properly conducted measurement of the value of si on a would find that value in ft just in case the value actually lies in ft (or, at least, would have lain in ft had the measurement not altered the value of si while measuring it, just as a thermometer might alter the temperature of a substance while taking it, an effect which in this case may be corrected for to yield the hypothetical undisturbed temperature of the substance). Perhaps the main problem for the naive realist comes from a set of arguments based on no-hidden-variable proofs.9 These seem to show that even if the precise values principle endorsed by the naive realist were true, it would be impossible to assign a value to every dynamical variable on each of a large number of similar systems in such a way that for each quantity, the fraction having each value is even close to the probability specified by the Born rules. It seems mathematically impossible to interpret the Born rules uniformly as 9

See, for example, Healey (1979), and Redhead (1987). The main no-hidden-variable results are contained in Gleason (1957) and Kochen and Specker (1967). The naive realist approach also has particular difficulties in accounting for violations of the Bell inequalities without postulating a kind of instantaneous action at a distance which is in conflict with the basic principles of relativity theory; see Chapter 5.

giving probability distributions over possessed values of dynamical variables. Since this claim continues to be disputed, and the arguments surrounding it are both complex and dependent on highly mathematical results, I cannot pursue the issue further here. But the naive realist approach has always been at most an interesting heretical alternative to the more orthodox Copenhagen viewpoint which I consider next. In the Copenhagen view, the Born rules explicitly concern the probabilities for various possible measurement results. They do not concern possessed values of dynamical variables. Indeed, according to this view, on each system there will always be some dynamical variables which do not possess precise values. In the Copenhagen interpretation, the Born rules assign probabilities to events of the form "The measured value of si on a lies in ft." Since the statement of the Born rules then involves explicit reference to measurement (or observation), to complete the interpretation it is necessary to say what constitutes a measurement. Proponents of the Copenhagen interpretation have typically either treated 'measurement' (or 'observation') or cognates as primitive terms in quantum mechanics, or else have taken each to refer vaguely to "suitable" interactions involving a "classical system." Each of these accounts is problematic. If "measurement" remains a primitive term, then it is natural to interpret it epistemologically as referring to an act of some observer which, if successful, gives him or her knowledge of some structural feature of a phenomenon. But then, quantum mechanics seems reduced to a tool for predicting what is likely to be observed in certain (not very precisely specified) circumstances, with nothing to say about the events in the world which are responsible for the results of those observations we make, and with no interesting implications for a world without observers. And indeed this instrumentalist/pragmatist conception of quantum mechanics has often gone along with the Copenhagen interpretation. On the other hand, if a measurement is a "suitable" interaction with a

"classical system," we need to know what interactions are suitable, and how there can be any "classical systems," if quantum mechanics is incompatible with and supersedes classical mechanics. In order to clarify and amplify these problems, it is useful to distinguish between two different versions of the Copenhagen interpretation. I suspect that whereas the first version is more familiar to many physicists, it is the second version which comes closer to representing Bohr's own view. In what I shall call the weak version of the Copenhagen interpretation, the dynamical properties of an individual quantum system are fully specified by means of its quantum state. A dynamical variable si possesses a precise real value a{ on a system if and only if that system is describable by a quantum state for which the Born rules assign probability one to the value a{ of si. In that state, a measurement of si would certainly yield the value a{. In other states, for which there is some chance that value a( would result if si were measured, and some chance that it would not, it is denied that si has any precise value prior to an actual measurement of it. Nevertheless, within the limits of experimental accuracy, measurement of a dynamical variable always yields a precise real value as its result, and this raises the question of the significance to be attributed to this value, given that it is typically not the value the variable possessed just before the measurement, nor the value it would have had if no measurement had taken place. One natural response is to say that the measured variable acquires the measured value as a result of the measurement: And then the Born rules explicitly concern the probabilities that dynamical variables acquire certain values upon measurement. Putting this response together with the condition for ascribing a precise real value to a variable given earlier, one concludes that after a precise measurement of a dynamical variable, a system is describable by a quantum state for which the Born rules assign probability one to the measured value of that variable. And this is one form of the projection postulate - a con10

troversial principle which purports to specify how the quantum state of a system changes as a result of the measurement of a dynamical variable on that system. In this popular version of the Copenhagen interpretation, a quantum system has a dynamical property if and only if it is describable by a quantum state for which the Born rules assign probability one to that property: A system has a property just in case observation would be certain to reveal that property. And for each individual quantum system there is a unique quantum state which in this way completely specifies the dynamical properties of that system. It follows that conjugate quantities such as (corresponding components of) position and momentum never have simultaneous precise values on any quantum system. The Heisenberg indeterminacy relations are then interpreted as limiting the precision with which (say) position and momentum are simultaneously defined on any individual quantum system. Since, in this version, measurement effects significant changes in the dynamical properties of a system, it is important for a proponent of the interpretation to specify in just what circumstances such changes occur. One might expect that such a specification would be forthcoming in purely quantum mechanical terms, through a quantum mechanical account of measuring interactions. Such an account would show how a physical interaction between one quantum system and another, which proceeds wholly in accordance with the principles of quantum mechanics, can effect a correlation between an initial value of the measured variable on one system (the "object" system) and afinal"recording property" on the other ("apparatus") system. The problem of giving such an account has become known as the quantum measurement problem. A solution to the measurement problem would explain the reference to measurement in the Born rules in purely physical (quantum mechanical) terms, and would also show to what extent the projection "postulate" may be considered a valid principle of quantum mechanics. Unfor11

tunately, despite much effort devoted to this task, it seems that such an account cannot be given within the constraints of this version of the Copenhagen interpretation: The measurement problem has proven intractable. Again, this is a complex and technical area which cannot be adequately reviewed here.10 But the key difficulty may be stated quite simply. It is that many initial states of an object system give rise to final compound object + apparatus quantum states which, in the present interpretation, imply that the apparatus fails to register any result at all (its pointer points nowhere at all)! For, in such afinalcompound quantum state, the Born rules do not assign probability one to any recording property of the apparatus system. The "paradox" of Schrodinger's cat provides a dramatic illustration of this difficulty. As is well known, Schrodinger (1935) described a thought experiment in which the state of a microscopic device (a radioactive nucleus) is coupled to a macroscopic system in such a way that if the nucleus has definitely decayed after a certain period of time a cat will definitely be dead, whereas if the nucleus has definitely not decayed after this time period, the cat will definitely be alive. The cat may be thought of as a rather unconventional (and indeed ethically questionable!) apparatus, whose condition records whether or not the nucleus has decayed. If the state of the nucleus at the beginning of the period is such that there is a 50 percent chance of it being observed to decay during the period, then, in the absence of any observation or other intervening interaction, at the end of the period the quantum state of the joint system nucleus + cat (-h other coupled intermediate systems) will correspond to the cat's being neither dead nor alive: though, of course, there would then be a 50 10

The classic presentation of the measurement problem is by Wigner (1963). That a more general quantum mechanical treatment fails to solve the problem is indicated, for example, by Fine (1970), whose proof is supplemented by Shimony (1974).

12

percent chance of observing the cat to be dead (an unusual variant on the maxim that it was curiosity that killed the cat!). The popular version of the Copenhagen interpretation faces the measurement problem because it takes each individual quantum system (for example, a single silver atom) to be describable by a quantum state which wholly determines its dynamical properties. But there is another version of the Copenhagen interpretation which views quantum states differently. According to what I shall call the strong version of the Copenhagen interpretation, it is mistaken to ascribe a quantum state to an individual system; a quantum state may be correctly ascribed only to an "ensemble" - that is, to a set of similar systems (such as a beam of silver atoms emerging from an oven), which share a certain physical history not possessed by a random collection of spatiotemporally dispersed similar systems. In this view, to ascribe a quantum state to an ensemble at a time is to say nothing about the dynamical properties of its elements at that time: Rather, the ascription of a quantum state to an ensemble is just a theoretical device which permits (correct) predictions concerning the statistics of experimental results, if the dynamical properties of members of the ensemble are observed. In this sense, the Born rules (together with techniques for describing particular ensembles by particular quantum states) exhaust the significance of the quantum state. Quantum mechanics simply has nothing to say about the dynamical properties of any quantum system at a time when it is not being observed.11 The weak version of the Copenhagen interpretation appeared to imply that a measuring apparatus may end up in a state in which it fails to have any well-defined property corresponding, for example, to the pointer position. And this implication conflicts with the belief we normally take to be "Indeed, in Bohr's own view, it would be meaningless to ascribe any dynamical property to a quantum system except in the context of a well-defined experimental arrangement suitable for the observation of that property.

13

warranted by our everyday observations, that no macroscopic apparatus ever fails to have some such directly perceptible property. However, in the strong version of the interpretation, one cannot apply quantum mechanics to an individual measurement interaction at all, but at most to an ensemble of similar such interactions. And even then, if one were to describe the ensemble of compound object + apparatus systems quantum mechanically, it would be illicit to infer anything about the final condition of the apparatus systems from the postinteraction quantum state of the ensemble. The restricted significance of the quantum state permits no such inference. The problem with the strong version does not arise because in certain circumstances a description in terms of the quantum state contradicts something we take ourselves to be justified in believing about the condition of a macroscopic apparatus after a measurement. It arises because this description never implies anything at all about that condition. On the one hand, the strong version of the Copenhagen interpretation takes the Born rules to yield probabilities for the possible results of measurements; whereas, on the other hand, in this view, a direct application of quantum mechanics to the compound object + apparatus systems involved in a measurement interaction fails to show that a measurement will yield any result at all. The view tries to solve this problem by requiring that the measuring apparatus be described classically. But what does this mean, and how does it help? It is plausible to suppose that to treat the measuring apparatus classically is to assume that it possesses a classical dynamical state, specified by a point in a classical phase space, and, perhaps, also that this state evolves in accordance with the laws of classical dynamics during the measurement interaction. That supposition would, of course, be highly problematic. Even though it may be macroscopic, the apparatus presumably consists of atoms; and it is the empirical inadequacy of classical mechanics as 14

applied to atomic systems which necessitated the transition to quantum mechanics. I suggest therefore that this interpretative supposition is wrong, and that, on the strong Copenhagen view, to give a classical description of the apparatus is not to assume that classical mechanics holds for it. Rather, it is to assume that the apparatus records a definite outcome of the measurement - that is, after the measuring interaction, the apparatus may be assigned a dynamical state which is classical in the minimal sense that the apparatus definitely has some enduring macroscopic property which can serve to record the outcome of the interaction. On the strong Copenhagen view, one cannot prove that the apparatus records a definite outcome from any description of the apparatus by means of a quantum state. But in the strong version of the Copenhagen interpretation (unlike the weak version) a definite measurement outcome is not inconsistent with any such description. And since no such inconsistency is threatened, the measurement problem does not arise in its usual form. Schrodinger's cat is definitely alive or dead before it is observed, but quantum mechanics cannot say which. Indeed, in this view, to apply quantum mechanics to the cat, considered as a measuring apparatus, one must simply assume that it ends up either alive or dead at the conclusion of the interaction, so that one cannot even show that it will end up one or the other. Why can we not just take this as an independent assumption, which is amply justified by our everyday observations of macroscopic objects such as cats? There would be no prima facie inconsistency involved in incorporating some such assumption into the strong version of the Copenhagen interpretation. But exactly what form could that assumption take? The problem is that it proves very difficult to give a precise general statement of such an assumption consistent with the rest of the interpretation. The assumption that all systems always have precise values for all dynamical variables would 15

contravene the Copenhagen understanding of the Heisenberg relations, which states that these relations impose limits on the precision with which conjugate variables are simultaneously defined on any system:12 it would also lead to many of the same problems faced by a naive realist interpretation. On the other hand, to give content to the weaker assumption that a measuring apparatus always emerges from a measurement interaction with some definite recording property at least requires a precise specification of what is to count as a measuring apparatus, a measurement interaction, and a recording property. But no such specification is offered. Moreover, one cannot simply assume that experimental physicists have developed clear operational tests for these notions, which might then be taken to provide at least a sort of working understanding of them. For if one considers the way in which quantum mechanics is applied, it becomes apparent that there is no unique correct way of dividing up any given experimental arrangement into object system and apparatus system. Many such divisions are possible, even though the results depend in detail on just how the division is made. All one can do is to apply a rule like that stated by Bell (1987): . . . . the following rule. . . , although ambiguous in principle, is sufficiently unambiguous for practical purposes: put sufficiently much into the [object] system that the inclusion of more would not significantly alter practical predictions, (p. 124)

We have now pushed the problem for the strong version of the Copenhagen interpretation back until it assumes the form in which Bell states it: The problem is this: quantum mechanics is fundamentally about "observations". It necessarily divides the world into two parts, a 12

Bohr himself would reject this assumption for the stronger reason that it countenances ascription of dynamical properties to quantum systems outside the context of any experimental arrangement suitable for their observation.

16

part which is observed, and a part which does the observing. The results depend in detail on just how this division is made, but no definite prescription for it is given. All we have is a recipe which, because of practical human limitations, is sufficiently unambiguous for practical purposes. So we may ask with Stapp: "How can a theory which is jundamentally a procedure by which gross macroscopic creatures, such as human beings, calculate predicted probabilities of what they will observe under macroscopically specified circumstances ever be claimed to be a complete description of physical reality?" (ibid., p. 124) Now Stapp's question (echoed by Bell in the last sentence of the quoted passage) actually raises a further problem for the Copenhagen interpretation, in either version. The problem arises as follows. Quantum mechanics is clearly a very successful theory with very wide applications. Moreover, it is commonly held to be not only predictively accurate, but also explanatorily powerful - perhaps even the most explanatorily powerful theory of contemporary physics. But if the Copenhagen interpretation were right, then, strictly, all that quantum mechanics describes, and permits one to predict, are the results of observations. Now we ordinarily assume that there is far more going on in the world than we are able to observe, and that the primary explanatory task of physics is to account for what actually occurs, whether or not we happen to observe it. Indeed, we ordinarily assume that a satisfactory explanation even of what we do observe will appeal to those hidden processes and mechanisms which are responsible for our observations. These assumptions support the charge that the Copenhagen interpretation of quantum mechanics is both descriptively and explanatorily incomplete. First, accepting this interpretation means accepting that quantum mechanics is descriptively incomplete on a massive scale. It does not describe the structure of elementary particles, nuclei, atoms, metals, superfluids, superconductors, lasers, crystalline solids, and semiconductors: It merely describes what we find when we observe such 17

things. Indeed, on this view, quantum mechanics would have nothing to say about a world in which no observations ever took place; and depending on exactly what is taken to be required for an observation, it may not even have anything to say about our world before there were entities such as humans capable of observing it. Second, the Copenhagen interpretation renders quantum mechanics explanatorily defective. Not only does it fail to explain phenomena that are not observed - including phenomena which we ordinarily take ourselves to have excellent reasons to suppose in fact occur, such as thermonuclear reactions in distant stars - but also the explanations it does offer of those phenomena that are observed amount to little more than predictions of their occurrence, with no independent accompanying account of whatever (perhaps microscopic) processes and events gave rise to them. Although this last problem for the Copenhagen interpretation may seem much more profound than those raised earlier, I have left it until last because proponents of the Copenhagen interpretation have a ready response to this problem, which draws heavily on certain general views of the nature and purposes of science, and the character of scientific explanation. The basic thrust of the response is to argue that the goal of science is to develop theories and techniques for successfully predicting our observations, that talk of any "reality" independent of our observations is metaphysical and unscientific, and consequently it is not part of science to describe such a "reality," and that, since explaining a phenomenon differs only psychologically from describing it as conforming to general regularities (insofar as we like an explanation to provide a "mental picture" of the phenomenon to be explained), we are bound to find quantum mechanical "explanations" unsatisfying just because they exclude the possibility of forming any "mental picture" of the phenomena. This response to the charge that the Copenhagen interpre18

tation would render quantum mechanics descriptively and explanatorily incomplete clearly rests on particular answers to central, though disputed, questions in the philosophy of science. An adequate rebuttal of the response would therefore require an extensive philosophical critique of these answers — something which has no place in this monograph. Therefore, I shall simply state my contrary convictions: that it is a primary goal of science to explain phenomena in the natural world; that our observations are only of interest insofar as they give us access to some of these phenomena; and that there are objective criteria for assessing the adequacy of a scientific explanation which have nothing to do with a psychological preference for mentally picturable models. These convictions are a primary motivation for the project I undertake in this book. Moreover, I suspect that it was similar convictions which motivated the development of quantum mechanics, and which continue to play a significant role in influencing people to devote themselves to those branches of contemporary physics which rest on this theory. It would be disappointing if the only way to understand much of contemporary physics were to reject beliefs which motivated one to make the attempt to do so. I turn now to a brief critical discussion of three unorthodox approaches to quantum mechanics: Each of these is motivated by dissatisfaction with the perceived incompleteness of the Copenhagen interpretation, and also by the perceived inadequacy of its treatment of the measurement process. Everett (1957) proposed a radical reinterpretation of quantum mechanics, and his proposal gave rise to what has become known as the many-worlds interpretation, though his own name ('relative state formulation') may in the end prove less misleading. According to this interpretation, all interactions, including measurement interactions, may be considered as quantum mechanical interactions internal to some compound system; there is no need to assume the existence of any classical systems; and the entire universe may itself be treated as 19

a single immensely complex quantum system, and assigned its own quantum state. Every other quantum system is treated as a subsystem of the universe. Interactions among such subsystems would soon prevent each from being described by its own unique quantum state: Nevertheless, each subsystem may always be assigned many relative (quantum) states, in such a way that there is a correlation between the relative states of any two systems which together compose the universe. Moreover, the dynamical properties of a subsystem in a particular relative state are just those which would be assigned probability one by application of the Born rules to that state. In a model universe composed just of an "object" system and an "apparatus" system, a quantum mechanically described measurement interaction would give rise to a quantum state for the compound system for which the Born rules do not assign probability one to any property corresponding to a definite measurement outcome. Nevertheless, each of the many apparatus relative states does assign probability one to some definite measurement outcome - though typically a different outcome for each such state. It is as though the pointer points everywhere at once, or Schrodinger's cat is both dead and alive! This raises two problems, one internal and one empirical. The internal problem is to show why it is not straightforwardly inconsistent to maintain that an "apparatus" system simultaneously possesses incompatible properties. One version of the many-worlds interpretation seeks to restore consistency by maintaining that measurement interactions induce a fission of the systems which undergo them: Before the interaction there is a single apparatus system, but after the interaction there are many, recording different results. This seems both ontologically extravagant, and observationally unwarranted, since we never observe such fissions. Instead, we always observe a single, definite result of any properly conducted quantum measurement, even though in the many20

worlds interpretation there are often supposed to be many different results: This is the empirical problem. Proponents of the interpretation have made strenuous and ingenious efforts to solve these problems, but in my opinion without success.13 However, despite its problems, the many-worlds interpretation is not without value, for its elegant treatment of many conceptual difficulties of quantum mechanics may provide models for the development of parallel solutions within an alternative interpretation like that offered in this book. Quantum logic is a confusing subject, and it is hard to pin down what is supposed to be involved in various attempts to give a quantum logical interpretation of quantum mechanics. Central to the interpretations of Putnam (1968) and Bub (1974) was the claim that quantum logic permits one to maintain the naive realist's understanding of dynamical states and their relation to the Born rules. Specifically, they held that the adoption of a nonclassical logic in the quantum domain enables one consistently to maintain that every quantum dynamical variable always has a precise real value, and that measurement is merely our way of getting to know that value, in the face of the arguments against this based on the no-hidden-variable proofs mentioned in footnote 9. Apart from general objections to the adoption of a nonclassical logic for empirical reasons,14 there are more specific objections to this particular proposal, viewed as an attempt to breathe new life into naive realism. The content of the claim that every dynamical variable has a precise value is quite obscure once one has adopted quantum logic, given that the standard (classical) inferences can no longer be drawn from it. And, in particular, it is by no means clear that the truth of this claim suffices to permit a naive realist reading of the Born rules, 13

See the references given in footnote 5, this chapter. See, for example, Dummett (1978), as well as Kripke's unpublished talk, "The Question of Logic," given at the University of Pittsburgh in 1974.

14

21

according to which the object system possessed the specific value revealed by a measurement prior to, or independent of, the occurrence of that measurement.15 Finally, I wish to briefly consider hidden variable theories. A clear motivation behind the construction of such theories has been the belief that some more complete account of microscopic processes is required than that provided by quantum mechanics according to the Copenhagen interpretation. The general idea has been to construct such an account by introducing additional quantities, over and above the usual quantum dynamical variables (such as de Broglie's pilot wave, Bohm's quantum potential, or fluctuations in Vigier's random ether), and additional dynamical laws governing these quantities and their coupling to the usual quantum variables. The primary object is to permit the construction of a detailed dynamical history of each individual quantum system which would underlie the statistical predictions of quantum mechanics concerning measurement results. Though it would be consistent with this aim for such dynamical histories to conform only to indeterministic laws, it has often been thought preferable to consider in the first instance deterministic hidden variable theories. A deterministic hidden variable theory would underlie the statistical predictions of quantum mechanics much as classical mechanics underlies the predictions of classical statistical mechanics. In both cases, the results of the statistical theory would be recoverable after averaging over ensembles of individual systems, provided that these ensembles are sufficiently "typical": but the statistical theory would give demonstrably incorrect predictions for certain "atypical" ensembles. Now, as Bell (1964) first showed, no deterministic hidden variable theory can reproduce the predictions of quantum mechanics for certain composite systems without violating a 15

For objections along these lines, see Putnam (1981), Stairs (1983), and Healey (1977).

22

principle of locality. And this principle seems firmly grounded in basic assumptions concerning the lack of physical connection between spatially distant components of such systems; and the impossibility of there being any such connection with the property that a change in the vicinity of one component should instantaneously produce a change in the behavior of the other. Further work attempting to extend Bell's result to apply to indeterministic hidden variable theories has shown that there may be a small loophole still open for the construction of such a theory compatible with the relativistic requirement that no event affects other events outside of its future light-cone.16 But to my knowledge no plausible theory has succeeded in exploiting this loophole, and I shall not seek to do so in this book. Existing hidden variable theories, such as that of Vigier (1982), are explicitly nonlocal, and do involve superluminal propagation of causal influence on individual quantum systems, although it is held that exploiting such influence to transmit information superluminally would be extremely difficult, if not actually impossible. Against this, it has often been maintained that any superluminal transmission of causal signals would be explicitly inconsistent with relativity theory: If this were so, such nonlocal hidden variable theories could be immediately rejected on this ground alone. But as I shall argue in Chapter 5, relativity does not explicitly forbid such transmission. Nonlocal hidden variable theories like that of Vigier can conform to the letter of relativity by introducing a preferred frame, that of the "subquantum ether," with respect to which superluminal propagation is taken to occur. By doing so they avoid the generation of so-called causal paradoxes. But they also thereby violate the spirit of relativity theory by reintroducing just the sort of privileged reference frame which it was Einstein's great achievement to have shown to be unnecessary for the formulation of (classical) 16

See especially Jarrett (1984).

23

mechanics and electromagnetism. Moreover, they do so while at the same time giving reasons as to why it would be at least extremely difficult to find out what this frame is. And this is why I find these theories so implausible. The principle that a fundamental theory can be given a relativistically invariant formulation seems so fundamental to contemporary physics that no acceptable interpretation of quantum mechanics should violate it.17 There is a more basic reason why I cannot accept a hidden variable theory as an interpretation of quantum mechanics. A hidden variable theory is, fundamentally, a separate and distinct theory from quantum mechanics. To offer such a theory is not to present an interpretation of quantum mechanics but to change the subject.18 Now it does follow from my characterization of what is involved in interpreting a theory that, prior to interpretation, the bounds of a theory are somewhat ill-defined. And my own interpretation will draw the bounds of quantum mechanics in a perhaps unusual way. But there seem to be clear reasons for denying that a hidden variable theory should count as part of quantum mechanics. One reason is that a hidden variable theory incorporates quantities additional to the quantum dynamical variables. Another is that hidden variable theories are held to underlie quantum mechanics in a way similar to that in which classical mechanics underlies the distinct theory of statistical mechanics. A final reason is that a hidden variable theory (at least typically) is held to be empirically equivalent to quantum me17

Of course, an interpretation of nonrelativistic quantum mechanics must portray that theory as violating relativistic invariance. But there should be nothing inherent in the interpretation itself which prevents its generalization to a fully relativistically invariant quantum mechanics. 18 This point may have been obscured by the approach, and especially by the title, of Bohm (1952a, 1952b). In these papers Bohm combined objections to the Copenhagen interpretation with the proposal of a hidden variable theory to underlie quantum mechanics, now interpreted according to his own avowedly nonCopenhagen approach. But his non-Copenhagen approach to quantum mechanics did not require the proposal of a hidden variable theory: It merely permitted this.

24

chanics only with respect to a restricted range of conceivable experiments, while leading to conflicting predictions concerning a range of possible further experiments which may, indeed, be extremely hard to actualize.

25

1 Overview No interpretation of quantum mechanics can be wholly comprehended or validated before carefully investigating all of its details. I begin such an investigation of the present interpretation in the next chapter. But some of these details will prove rather intricate. The reader may need help in seeing the forest formed by the trees, and the skeptical reader may require encouragement before he or she agrees to enter its thickets. In this first chapter I therefore present a preliminary account of the basic ideas of the interpretation to be given, which may serve as a guide to what is to follow. Just as parts of a travel guide may seem puzzling before one actually visits the region described, so also parts of this chapter will likely seem much clearer when they are reread in the light of the rest of the book. There is another reason for beginning with a general overview. The present interpretation arose as what seemed to be the most promising way of implementing a certain general approach to quantum theory. But, though I am confident of the worth of this approach, the present interpretation may well come to be regarded as only a fruitful first attempt at its detailed implementation, requiring more or less extensive technical revision if it is to prove ultimately satisfactory. If so, this overview may continue to be useful as a guide to an interpretation whose technical framework differs more or less radically from that underlying the present interpretation. Let me start with the concept of the state of a system. In quantum mechanics this is normally taken to be represented 26

by a mathematical object, such as a wave function, state vector, or density operator. Although this remains broadly true in the present approach, it becomes important to distinguish between two different conceptions of the state of a system: the dynamical state and the quantum state. In most interpretations, including the present one, the primary role of the quantum state is to generate probabilities, concerning the possible outcomes of measurements, via the Born rules. Thus, if a system is ascribed quantum state i|/, then the probability that a measurement of dynamical variable si would give a value lying in (Borel set) 11 is written as prob^, (s#Eft), where this probability is calculated according to the appropriate quantum algorithm [for example, prob^ (^JEll) = (v|;,P • • • Ha». If a is neither null nor atomic, then a key feature of the present interpretation is the division of the set of all possible dynamical properties of a into two subclasses. A dynamical property of a is composite just in case its corresponding projection operator PCT is expressible as a direct product P2) if and only if VU 2 *. • (X7X'F), where each expansion is biorthogonal, and there is some si* such that, for all IEJ, A'xf = 4x7, A'xT = a-x'7. Then, if it is true for the system representative of a before the interaction that si = ai9 it is also true for the system representative of a after the interaction. In general, however, if a system is subject to external interactions, then its system representative may change indeterministically. The present interpretation of quantum mechanics is silent on the details of such indeterministic transitions, except to assume that their probabilities vary with the strength of the interaction, becoming negligible as the interaction strength approaches zero. In the present interpretation, only the cumulative effect of these indeterministic transitions is specified by quantum mechanics; and this is to leave a system at the conclusion of an interaction with a 82

system representative from the privileged set, whose probability is given by the subspace decomposition condition. Moreover, in the present interpretation, these probabilities for being left with a particular system representative at the conclusion of an interaction are the ultimate source of all quantum mechanical probabilities. This may seem surprising, since quantum mechanics is usually postulated to yield probabilities of measured results on a system in a given quantum state, and neither measurement nor quantum states have yet been mentioned. It will still hold for the present interpretation that quantum mechanics yields probabilities of measurement results on systems in a given quantum state. However, this is not secured by postulation, but must instead be derived through an analysis of the nature of a measurement interaction and the conditions under which quantum states may be assigned to systems. Such analysis occupies the next chapter: Only at its conclusion will the traditional quantum probabilistic algorithm emerge.

83

Measurement and quantum states §3.1

MEASUREMENT INTERACTIONS

In the present interpretation terms such as 'measurement,' 'observation,' 'find,' 'classical,' and 'apparatus' do not appear in the formulation of quantum mechanics. Yet it must of course be true that quantum mechanics can be used to predict features of the results of measurements - in particular their statistics. What is required, then, is an account in quantum mechanical terms of those physical interactions capable of being used to perform measurements of quantum dynamical variables, from which it follows that the predicted probabilities for specified behavior of certain quantum systems in these interactions coincide with what, on a more conventional interpretation, are simply postulated as the quantum mechanical probabilities for various measurement results on a system with a given quantum state. It is necessary to see how this consequence may be derived, not only in order to explicate the Born rules, but also to explain the role of quantum states within the present interpretation. The first step toward the required account is to analyze the nature of an interaction capable of being used to perform a measurement of the value of a quantum dynamical variable in a quantum system. Any such interaction will be called an Af-type interaction, whether or not it is in fact used to perform a measurement. The basic characteristic of an M-type interaction is that it establishes a correlation between the initial dynamical state of one quantum system a (the object system) and the final dynamical state of another quantum system a (the apparatus 84

system). Though a and a will usually be distinct systems, in some cases an M-type interaction may correlate initial and final dynamical states of a single system, for example, by coupling its prior spin to its subsequent position. What kind of interaction is capable of establishing the needed correlation? One could view this question as a request for a single, precise definition of the class of M-type interactions. But though I shall offer definitions of certain idealized kinds of M-type interaction, there is no reason to suppose that these exhaust the entire class of M-type interactions. In the present interpretation, that class may be fairly heterogeneous, its various subclasses having only two central features in common: the ability to set up a suitable correlation between dynamical states, and the fact that interactions in all these subclasses support the ascription of a quantum state to one interacting system and its application via the Born rules. Although it may be of some interest to conduct a systematic theoretical investigation of the entire class of M-type interactions, I shall not do so in this monograph. What I shall do is to show that there can be at least some M-type interactions, by precisely characterizing one simple variety of M-type interaction and then showing how interactions of this type would indeed establish the right kind of correlation between dynamical states, and would also support the ascription of a quantum state and its use in the Born rules. What I shall call a simple M-type interaction involving a system a proceeds in accordance with a characteristic interaction Hamiltonian linking a to a suitable M-type system a (which may consequently be thought of as an apparatus for the measurement). To say when a simple M-type interaction involving a occurs, it is therefore necessary to specify the nature of this interaction Hamiltonian, the general features of an M-type system, and the required condition of the particular M-type system a at the start of its interaction with a. An interaction between two quantum systems a and a is M-suitable for si just in case it is characterized by an inter85

action term H7 in the universal Hamiltonian which is zero outside an interval (t(htA), a n d would produce the following Schrodinger evolution in a (hypothetical) compound quantum state vector of a 0 a over the interval (t(htx), for each element in some orthonormal basis {x//} of vectors for Ha with Ax^ = d,X//> a n d some subspace P" of vectors in H",

xJ®xS-^€J®x^

(3-1)

where ( g , ^ ) = (x*,X?/) = 8,, • 8^, Xo^Po- Note that it is not assumed that a may be ascribed some quantum state vector (either \7j or anything else) at t0: Equation (3.1) is simply a (rather weak) mathematical restriction on the form ofH 7 . For an M-type interaction to actually occur, it is not enough that a pair of systems interact via an M-suitable interaction: One of these systems must be an M-type system which is M-ready at the start of the interaction. A system is M-ready for si at t if and only if its system representative at t is a vector Xo from the subspace PJJ for which the interaction is M-suitable for si. Suppose that an interaction occurs between a and a which is M-suitable for si. Let P" < H" be defined as [x?/. i fixed]: 2P" is the recording property of a corresponding to the value a{ of si on a. (Here and elsewhere the notation [ ] indicates the subspace spanned by the vectors in brackets.) The value a{ of si is manifested in an M-suitable interaction just in case a has the recording property 2?" at the conclusion of the interaction. Then a is an M-type system if and only if i. the SP? are all prime properties of a, and ii. there is no nontrivial composite property 2" of a such that, for some i, " < a a . [Note that (i) is actually redundant, since it follows from (ii): I am indebted to Howard Stein for pointing this out to me. ] Finally, a simple M-type interaction (SMI) for si on a occurs between t0 and tx if and only if a and a interact through 86

an interaction which is M-suitable for si, where a is an Mtype system which is M-ready for si at t(). It follows that distinct recording properties 2P", 2P" are noncotenable. Hence, any set of distinct real numbers {q} defines the possible values of a recording quantity 21 which corresponds to the self-adjoint operator Q a = 2#,-P?: 21 has value qt on a if and only if a has recording property SP". Note that though Equation (3.1) constrains H 7 , this is a constraint which can be met: In particular, the arrow represents unitary evolution over the interval (tihtx). Note also that any atomic system trivially counts as an M-type system, but no real apparatus is an atomic system. To see why an interaction meeting these conditions is capable of being used to perform a measurement of the value of a quantum dynamical variable, consider the effect of such an interaction on a system cr when the initial universal system representative has the form r( i = i\ it follows from the subspace decomposition condition that the final system representatives of a and a are the pair ([£//], [X//])> f° r some /, where j varies over some index set Ia such that for all j,j' E Iih \cij\2=\cij"\2. For later use, set U//,7 = /,. Consider first the case in which ^- = 0 for i^k. Let fT(s& = ak) be the property which a has when si = ak. If ^(si = ak) is an irreducible property at t(h then as long as the proviso is met, a initially has 87

2PCT(,9# = ak) by the system representative condition. And since a is an M-type system, it follows that 3P£ is prime, and that there is no composite property weaker than 2P£ that a does not have; the system representative condition consequently implies that a has 9*£ at tx. Hence, if the initial universal system representative is ^(fo), and si initially has value ak on a, then a is certain to acquire the recording property 2P£ as a result of the interaction. Consider now the case in which there are distinct i,i' with c^O, c^^O (for somej,/). Initially, a does not have 2Pa(s4 = a) for any i. For since this is assumed to be an irreducible property of a at t(), a could have it only by virtue of the system representative condition, which it does not have. Hence, si does not initially have a precise value on a. Nevertheless, the final system representative of a is [x£: fixed i], with probability 2>G/;/ \c{j\2. Consequently, for some i, a has 9*" at f,. Thus, in this case also a acquires some recording property 9*" as a result of the interaction: Indeed, the probability that it acquires 2Pf is %j(EIi \c{j\2. This is why an SMI gives a definite result, even when the measured variable had no prior value. Note that any dynamical variable si measurable in an SMI must correspond to a self-adjoint operator with pure point spectrum, since its eigenvectors span Ha. Note also that a set of dynamical variables {sip} measurable simultaneously in the same SMI must be compatible; that is, their corresponding self-adjoint operators must all commute pair wise, or, equivalently, they must all share a complete set of eigenvectors. To show this, consider the effect of an SMI for any pair si, Si of quantities in this set. Any initial universal system representative *\faa(t0) may be expanded in each of two ways:

> X" =

where Ax^ =